Found problems: 966
1978 Putnam, B1
Find the area of a convex octagon that is inscribed in a circle and has four consecutive sides of length $3$ and
the remaining four sides of length $2$. Give the answer in the form $r+s\sqrt{t}$ with $r,s, t$ positive integers.
1989 Putnam, A3
Prove that all roots of $ 11z^{10} \plus{} 10iz^9 \plus{} 10iz \minus{}11 \equal{} 0$ have unit modulus (or equivalent $ |z| \equal{} 1$).
1970 Putnam, A5
Determine the radius of the largest circle which can lie on the ellipsoid
$$\frac{x^2 }{a^2 } +\frac{ y^2 }{b^2 } +\frac{z^2 }{c^2 }=1 \;\;\;\; (a>b>c).$$
1948 Putnam, A5
If $\xi_1,\ldots,\xi_n$ denote the $n$-th roots of unity, evaluate
$$\prod_{1\leq i<j \leq n} (\xi_{i}-\xi_j )^2 .$$
1988 Putnam, B4
Prove that if $\sum_{n=1}^\infty a_n$ is a convergent series of positive real numbers, then so is $\sum_{n=1}^\infty (a_n)^{n/(n+1)}$.
1962 Putnam, A6
Let $S$ be a set of rational numbers such that whenever $a$ and $b$ are members of $S$, so are $ab$ and $a+b$, and having the property that for every rational number $r$ exactly one of the following three statements is true:
$$r\in S,\;\; -r\in S,\;\;r =0.$$
Prove that $S$ is the set of all positive rational numbers.
1958 November Putnam, A6
Let $a(x)$ and $b(x)$ be continuous functions on $[0,1]$ and let $0 \leq a(x) \leq a <1$ on that range. Under what other conditions (if any) is the solution of the equation for $u,$
$$ u= \max_{0 \leq x \leq 1} b(x) +a(x)u$$
given by
$$u = \max_{0 \leq x \leq 1} \frac{b(x)}{1-a(x)}.$$
2022 Putnam, A2
Let $n$ be an integer with $n\geq 2.$ Over all real polynomials $p(x)$ of degree $n,$ what is the largest possible number of negative coefficients of $p(x)^2?$
1978 Putnam, A4
A [i]bypass[/i] operation on a set $S$ is a mapping $B: S\times S \rightarrow S$ with the property $B(B(w, x), B(y,z)) = B(w,z)$ for all $w, x, y, z \in S$.
(a) Prove that $B(a,b)=c$ implies $B(c,c)=c$ when $B$ is a bypass.
(b) Prove that $B(a,b)=c$ implies $B(a,x)=B(c,x)$ for all $x\in S$ when $B$ is a bypass.
(c) Construct a bypass operation $B$ on a finite set S with the following three properties
[list=i]
[*] $B(x,x)=x$ for all $x\in S$.
[*] There exist $d$ and $e$ in $S$ with $B(d,e)=d \ne e.$
[*] There exist $f$ and $g$ in $S$ with $B(f,g)\ne f.$
[/list]
2006 Putnam, A3
Let $1,2,3,\dots,2005,2006,2007,2009,2012,2016,\dots$ be a sequence defined by $x_{k}=k$ for $k=1,2\dots,2006$ and $x_{k+1}=x_{k}+x_{k-2005}$ for $k\ge 2006.$ Show that the sequence has 2005 consecutive terms each divisible by 2006.
1975 Putnam, A4
Let $m>1$ be an odd integer. Let $n=2m$ and $\theta=e^{2\pi i\slash n}$. Find integers $a_{1},\ldots,a_{k}$ such that
$\sum_{i=1}^{k}a_{i}\theta^{i}=\frac{1}{1-\theta}$.
1948 Putnam, B1
Let $f(x)$ be a cubic polynomial with roots $x_1 , x_2$ and $x_3.$ Assume that $f(2x)$ is divisible by $f'(x)$ and compute the ratio $x_1 : x_2: x_3 .$
2011 Putnam, A3
Find a real number $c$ and a positive number $L$ for which
\[\lim_{r\to\infty}\frac{r^c\int_0^{\pi/2}x^r\sin x\,dx}{\int_0^{\pi/2}x^r\cos x\,dx}=L.\]
2021 Putnam, B3
Let $h(x,y)$ be a real-valued function that is twice continuously differentiable throughout $\mathbb{R}^2$, and define
\[
\rho (x,y)=yh_x -xh_y .
\]
Prove or disprove: For any positive constants $d$ and $r$ with $d>r$, there is a circle $S$ of radius $r$ whose center is a distance $d$ away from the origin such that the integral of $\rho$ over the interior of $S$ is zero.
1981 Putnam, A4
A point $P$ moves inside a unit square in a straight line at unit speed. When it meets a corner it escapes. When it
meets an edge its line of motion is reflected so that the angle of incidence equals the angle of reflection.
Let $N( t)$ be the number of starting directions from a fixed interior point $P_0$ for which $P$ escapes within $t$ units of time. Find the least constant $a$ for which constants $b$ and $c$ exist such that
$$N(t) \leq at^2 +bt+c$$
for all $t>0$ and all initial points $P_0 .$
1959 Putnam, B3
Give an example of a continuous real-valued function $f$ form $[0,1]$ to $[0,1]$ which takes on every value in $[0,1]$ an infinite number of times.
2009 Putnam, B6
Prove that for every positive integer $ n,$ there is a sequence of integers $ a_0,a_1,\dots,a_{2009}$ with $ a_0\equal{}0$ and $ a_{2009}\equal{}n$ such that each term after $ a_0$ is either an earlier term plus $ 2^k$ for some nonnnegative integer $ k,$ or of the form $ b\mod{c}$ for some earlier positive terms $ b$ and $ c.$ [Here $ b\mod{c}$ denotes the remainder when $ b$ is divided by $ c,$ so $ 0\le(b\mod{c})<c.$]
2005 Federal Competition For Advanced Students, Part 2, 3
Let $Q$ be a point inside a cube. Prove that there are infinitely many lines $l$ so that $AQ=BQ$ where $A$ and $B$ are the two points of intersection of $l$ and the surface of the cube.
1960 Putnam, A4
Given two points, $P$ and $Q$, on the same side of a line $L$, the problem is to find a third point $R$ so that $PR+ RQ+RS$ is minimal, where $S$ is the unique point on $L$ such that $RS$ is perpendicular to $L.$ Consider all cases.
1997 Putnam, 1
A rectangle, $HOMF$, has sides $HO=11$ and $OM=5$. A triangle $\Delta ABC$ has $H$ as orthocentre, $O$ as circumcentre, $M$ be the midpoint of $BC$, $F$ is the feet of altitude from $A$. What is the length of $BC$ ?
[asy]
unitsize(0.3 cm);
pair F, H, M, O;
F = (0,0);
H = (0,5);
O = (11,5);
M = (11,0);
draw(H--O--M--F--cycle);
label("$F$", F, SW);
label("$H$", H, NW);
label("$M$", M, SE);
label("$O$", O, NE);
[/asy]
PEN A Problems, 14
Let $n$ be an integer with $n \ge 2$. Show that $n$ does not divide $2^{n}-1$.
1972 Putnam, B4
Show that for $n > 1$ we can find a polynomial $P(a, b, c)$ with integer coefficients such that
$$P(x^{n},x^{n+1},x+x^{n+2})=x.$$
1956 Putnam, B3
A sphere is inscribed in a tetrahedron and each point of contact of the sphere with the four faces is joined to the vertices of the face containing the point. Show that the four sets of three angles so formed are identical.
2010 Putnam, B5
Is there a strictly increasing function $f:\mathbb{R}\to\mathbb{R}$ such that $f'(x)=f(f(x))$ for all $x?$
PEN G Problems, 24
Let $\{a_{n}\}_{n \ge 1}$ be a sequence of positive numbers such that \[a_{n+1}^{2}= a_{n}+1, \;\; n \in \mathbb{N}.\] Show that the sequence contains an irrational number.